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Find the domain and range of the range of each of the following real functions: `f(x)=(1)/(sqrt(x^(2)-1))` |
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Answer» Correct Answer - dom `(f)=(-oo,-1)uu(1,oo)," range "(f)=R-{0}` `f(x)=(1)/(sqrt(x^(2)-1))` is defined only when `x^(2)-1gt0`. Now, `x^(2)-xgt0implies(x+1)(x-1)gt0implies(xlt-1)or(xgt1)` `impliesx""in(-oo,-1)orx""in(1,oo)`. `:."dom "(f)=(-oo,-1)uu(1,oo)`. Let y=f(x). Then, `y=(1)/(sqrt(x^(2)-1))impliesy^(2)=(1)/((x^(2)-1))impliesx^(2)-1=(1)/(y^(2))impliesx=sqrt((1)/(y^(2))+1)=sqrt((1+y^(2))/(y^(2)))` Clearly, x is not defined when y=0. `:."range "(f)=R-{0}`. |
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